Conjecture on strictly bounded objects in symmetric tensor categories

Let $\mathcal{C}$ be a symmetric tensor category over a field $\mathbf{k}$ of characteristic $p>0$. Let $X\in\mathcal{C}$ be strictly $N$-bounded, with $N>3$.

Strict boundedness conjecture. The following assertions hold:

1. Either $\operatorname{Sym}^2 X$ or $\wedge^2 X$ is invertible;
2. $N=p^n$ for some $n\in\mathbb{Z}_{>0}$;
3. $\operatorname{dim} X=\pm 2$;
4. $X$ is homotopically $N$-bounded.

This conjecture describes the expected structure of strictly bounded objects in positive-characteristic symmetric tensor categories. The supplied passage says that it will be proved for quasi-finite symmetric tensor categories, so the general statement remains open in the context provided.